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Register a new userA polynomial approach to the Collatz conjecture
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Added by
Feng Pan
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak proof of the conjecture by using induction with respect to the degree of the polynomials.
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We present a formulation of the Collatz conjecture that is potentially more amenable to modeling and analysis by automated termination checking tools.
Assuming Schanuels conjecture, we prove that any polynomial exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial exponential equations have only finitely many rational solutions. This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one-variable case of Zilbers strong exponential-algebraic closedness conjecture can be reduced to Schanuels conjecture.
Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the Collatz conjecture based on bit representation of integers. The scheme neither encounters any cycles in the so called Collatz sequence and nor the sequence grows indefinitely. Experimental results show that the Collatz sequence starting at the given integer , oscillates for finite number of times, never exceeds 1.7 times (scaling factor) size of the starting integer and finally reaches the value 1. The experimental results show strong evidence that conjecture is correct and paves a way for theoretical proof.
We present a variation of the modular algorithm for computing the Hermite normal form of an $mathcal O_K$-module presented by Cohen, where $mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$.
The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture.
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